Humans can walk without falling despite some external perturbations, but the control mechanisms by which this stability is achieved have not been fully characterized. While numerous walking simulations and robots have been constructed, no full-state walking controller for even a simple model of walking has been derived from human walking data. Here, to construct such a feedback controller, we applied thousands of unforeseen perturbations to subjects walking on a treadmill and collected data describing their recovery to normal walking. Using these data, we derived a linear controller to make the classical inverted pendulum model of walking respond to perturbations like a human. The walking model consists of a point-mass with two massless legs and can be controlled only through the appropriate placement of the foot and the push-off impulse applied along the trailing leg. We derived how this foot placement and push-off impulse are modulated in response to upper-body perturbations in various directions. This feedback-controlled biped recovers from perturbations in a manner qualitatively similar to human recovery. The biped can recover from perturbations over twenty times larger than deviations experienced during normal walking and the biped’s stability is robust to uncertainties, specifically, large changes in body and feedback parameters.

ResearchCite this article: Joshi V, Srinivasan M. 2018 Walking crowds on a shaky surface: stable walkers discover Millennium Bridge oscillations with and without pedestrian synchrony. Biol. Lett. 14: 20180564. http://dx.Why did the London Millennium Bridge shake when there was a big enough crowd walking on it? What features of human walking dynamics when coupled to a shaky surface produce such shaking? Here, we use a simple biped model capable of walking stably in three dimensions to examine these questions. We simulate multiple such stable bipeds walking simultaneously on a bridge, showing that they naturally synchronize under certain conditions, but that synchronization is not required to shake the bridge. Under such shaking conditions, the simulated walkers increase their step widths and expend more metabolic energy than when the bridge does not shake. We also find that such bipeds can walk stably on externally shaken treadmills, synchronizing with the treadmill motion for a range of oscillation amplitudes and frequencies. Our simulations illustrate how interactions between (idealized) bipeds through the walking surface can produce emergent collective behaviour that may not be exhibited by just a single biped.

Understanding how humans walk on a surface that can move might provide insights into, for instance, whether walking humans prioritize energy use or stability. Here, motivated by the famous human-driven oscillations observed in the London Millennium Bridge, we introduce a minimal mathematical model of a biped, walking on a platform (bridge or treadmill) capable of lateral movement. This biped model consists of a pointmass upper body with legs that can exert force and perform mechanical work on the upper body. Using numerical optimization, we obtain energyoptimal walking motions for this biped, deriving the periodic body and platform motions that minimize a simple metabolic energy cost. When the platform has an externally imposed sinusoidal displacement of appropriate frequency and amplitude, we predict that body motion entrained to platform motion consumes less energy than walking on a fixed surface. When the platform has finite inertia, a massspring-damper with similar parameters to the Millennium Bridge, we show that the optimal biped walking motion sustains a large lateral platform oscillation when sufficiently many people walk on the bridge. Here, the biped model reduces walking metabolic cost by storing and recovering energy from the platform, demonstrating energy benefits for two features observed for walking on the Millennium Bridge: crowd synchrony and large lateral oscillations.

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Individuals post-stroke experience persisting gait deficits due to altered joint mechanics, known clinically as spasticity, hypertonia, and paresis. In engineering, these concepts are described as stiffness and damping, or collectively as joint mechanical impedance, when considered with limb inertia. Typical clinical assessments of these properties are obtained while the patient is at rest using qualitative measures, and the link between the assessments and functional outcomes and mobility is unclear. In this study we quantify ankle mechanical impedance dynamically during walking in individuals post-stroke and in age-speed matched control subjects, and examine the relationships between mechanical impedance and clinical measures of mobility and impairment. Perturbations were applied to the ankle joint during the stance phase of walking, and least-squares system identification techniques were used to estimate mechanical impedance. Stiffness of the paretic ankle was decreased during mid-stance when compared to the non-paretic side; a change independent of muscle activity. Inter-limb differences in ankle joint damping, but not joint stiffness or passive clinical assessments, strongly predicted walking speed and distance. This work provides the first insights into how stroke alters joint mechanical impedance during walking, as well as how these changes relate to existing outcome measures. Our results inform clinical care, suggesting a focus on correcting stance phase mechanics could potentially improve mobility of chronic stroke survivors.